copyright 1996 by the american psychological association, inc....

Journal of Experimental Psychology:Human Perception and Performance1996, Vol. 22, No. 3, 531-543

Copyright 1996 by the American Psychological Association, Inc.0096-1523«6/J3.00

Do Fielders Know Where to Go to Catch the Ball or Only Howto Get There?

Peter McLeodOxford University

Zoltan DienesSussex University

Skilled fielders were filmed as they ran backward or forward to catch balls projected towardthem from a bowling machine 45 m away. They ran at a speed that kept the acceleration ofthe tangent of the angle of elevation of gaze to the ball at 0. This algorithm does not tellfielders where or when the ball will land, but it ensures that they run through the place wherethe ball drops to catch height at the precise moment that the ball arrives there. The algorithmleads to interception of the ball irrespective of the effect of wind resistance on the trajectoryof the ball.

The everyday nature of the act of running to catch a ballcan obscure the remarkable predictive ability that it re-quires. Figure 1 shows the trajectories of three balls pro-jected at 45° and approximately 22, 24, and 26 m/s towarda stationary fielder 45 m away. They will land 5 m in frontof, at, or 5 m behind the fielder, respectively. The solid lineshows the trajectory of each ball in the first 840 ms; thedashed line shows the rest of the flight. Within 840 ms, mostcompetent fielders would have started running forward forthe ball on the lower trajectory and backward for the ball onthe higher trajectory.1 Yet, the only difference betweenthese two flights at this time is the difference between thelongest and shortest solid lines. How is the fielder able towork out where to go from so little information?

Precise calculation of the trajectory is not possible be-cause the essential ball flight parameters of projection angle,velocity, and wind resistance are available to the fielderonly as, at best, crude estimates. Nor, given the infinitevariation of trajectory, does it seem possible that learning tocatch involves learning individual trajectories. An alterna-tive is that an algorithm exists that links the visual infor-mation obtained from watching the ball's flight to a runningspeed that will bring the fielders to the correct place, irre-spective of their starting position or the ball's trajectory.Learning to catch would involve the discovery of thisalgorithm.

This research was supported by a grant from the Oxford Uni-versity Research and Equipment Fund.

We are indebted to Tommy McLeod for some of the crucialobservations about the power of d2(tan a)/dt2. We thank ClareMendham and Daniel Chambers for analyzing the films. We aregrateful to a number of referees, particularly Michael McBeath, fortheir comments on earlier versions of this article.

Correspondence concerning this article should be addressed toPeter McLeod, Department of Experimental Psychology, OxfordUniversity, South Parks Road, Oxford OX1 3UD England, or toZoltan Dienes, Experimental Psychology, Sussex University,Brighton BN1 9QG England. Electronic mail may be sent viaInternet to [email protected] or [email protected].

Chapman (1968) analyzed the visual information avail-able to a fielder watching a ball approaching in parabolicflight. He showed that if a is the angle of elevation of gazefrom the fielder to the ball, then the acceleration of thetangent of a, d2(tan a)/dt2, will be zero if, and only if, thefielder is standing at the place where the ball will land. Healso suggested that this might be the basis of an interceptionalgorithm. A fielder who starts at a place other than wherethe ball will land and runs at a constant velocity that keepsd2(tan a)/dt2 at zero will arrive at the correct place to makethe catch at the same time as the ball.

However, because Chapman's (1968) work is based onthe information provided by watching an object in parabolicflight, it is not clear what relevance it has to catching. Windresistance ensures that objects in the real world do notfollow parabolic trajectories. The departure from parabolicflight can be substantial at the speeds encountered in ballgames. For example, Brancazio (1985) estimated that theeffect of wind resistance on a well-hit baseball would be toreduce the horizontal distance traveled by up to 40% of thedistance it would have achieved in parabolic flight. Further-more, given identical projection angle and initial velocity,different objects follow different trajectories because oftheir different wind resistances. If catching involves learn-ing an algorithm that links visual information to runningspeed, it must be one that works independently of the effectof wind resistance on trajectory.

Given Brancazio's (1985) analysis, one might be temptedto think that demonstrations of geometrical relationshipsthat could form the basis of algorithms for intercepting ballsin parabolic flight have no relation to real catching. How-ever, a recent article has suggested otherwise. Michaels andOudejans (1992) filmed two people running backward orforward to catch a ball. From the position of the catcher's

1 In the current study, for example, the fielder started running inthe correct direction within 840 ms for 74% of catches. Michaelsand Oudejans (1992) also found that catchers started to run shortlyafter the ball appeared.

531

532 McLEOD AND DIENES

First 840ms of flight

Subsequent flight15m

10m

t _ A

3

10m 20m 30m 40m 50m

Figure 1. The trajectories of three balls projected at 45° and a velocity (v) of 22.3, 24.0, and 25.7m/s, respectively, toward a fielder 45 m away. They experienced a deceleration due to aerodynamicdrag proportional to v2. The constant of proportionality was 0.007 m"1, a value typical of objectssuch as cricket balls (Daish, 1972).

head and of the ball, they were able to calculate the opticheight of the ball throughout its flight. (The optic height isthe position of the ball's image on an imaginary plane afixed distance in front of the fielder's eye.) They showedthat when fielders moved to make a catch, optic heightincreased with roughly constant velocity until just beforethe catch. Optic height is equivalent to the tangent of theangle of gaze, and constant velocity implies zero accelera-tion. Therefore, their result appears to offer support forChapman's (1968) proposal that interception is ensured byrunning at a speed that maintains the acceleration of thetangent of the angle of gaze at zero.2

Before one concludes that Chapman (1968) was correct, wemust elaborate on Michaels and Oudejans's (1992) result.First, their main experiment presented data from only 10 catch-es: 7 from one fielder and 3 from another. Second, they offeredno statistical test of the linearity of the plots of optic heightagainst time (i.e., of the claim that optic height increases atconstant velocity). Third, although it is possible to fit a straightline by eye to the early parts of the plots of optic height againsttime for each catch, in the majority of catches there is adeparture from linearity in the second half of the flight.3 Theydid not show whether Chapman's strategy will lead to inter-ception (and that these deviations are unimportant) or whetherthe deviations are a necessary corrective process becauseChapman's strategy does not actually get the fielder closeenough to the ball to catch it (because of the effects of windresistance). Finally, Michaels and Oudejans did not analyze thefielders' running velocity. Chapman's analysis requires notonly that fielders should keep the velocity of optic heightconstant as they run but also that they find the constant runningvelocity at which this happens. If Chapman's analysis explainshow fielders get to the right place at the right time, thiscondition must be met too.

The aim of the experiments reported here was to extendMichaels and Oudejans's (1992) analysis of whether run-ning speed is controlled by an algorithm linked to somefunction of the angle of elevation of gaze to cover the four

points above. We measured the running speed and the angleof elevation of gaze as skilled fielders ran to catch a ball.Successful interception usually requires the fielder to judgewhether the ball is going to the left or the right as well aswhether it is going to drop in front or behind. Visual cuesthat are available to make the left-right judgment have beenidentified (Regan, Beverley, & Cynader, 1979; see alsoRegan, 1993; Regan & Kanshal, 1994). Like Michaels andOudejans, we considered the remaining problem of whetherthe fielder should move backward or forward to catch theball. For simplicity, all our experiments involved balls pro-jected in a vertical plane between the point of projection andthe fielder so that the fielder did not have to move left orright. The algorithm that we show that fielders use in thissituation works equally well in the more general case wherethe fielder must decide whether to move left or right as wellas backward or forward.

Experiment 1

Method

Participants

Six skillful ball catchers participated. One was a professionalsoccer player, 1 played cricket at the professional level, and theremaining 4 were keen amateur cricket players. All were male.

2 Optic height and the tangent of the angle of gaze are mathe-matically equivalent quantities, so the choice of one rather than theother may seem arbitrary. Angle of gaze is available directly to afielder who looks at the ball; optic height is available directly to afielder who maintains fixation on the point from which the ballwas projected. Because fielders look at the ball when trying tocatch it, not at the point of projection, it seems more appropriate tochoose a function of the angle of gaze as the basis for the analysis.

3 There is also a catastrophic departure, just before the catch,that occurs too late to be relevant to the question of how the fielderarrives at the right place to make the catch.

RUNNING TO CATCH THE BALL 533

Catching

The fielder stood approximately 45 m from a bowling machine,which projected a hard white ball into the air directly toward himat a projection angle of 45°. For different deliveries the speed wasvaried randomly over a range of about 20-25 m/s so that the ballwould unpredictably go over his head or fall short, with a range ofabout 0 m around his starting position. About 50 balls werefired at each fielder. He ran backward or forward or stayed wherehe was, trying to catch each ball.

Measurement

Fielder's position and velocity. Figure 2 shows a bird's-eyeview of the experimental setup. The fielder ran backward andforward along an imaginary line between himself and the bowlingmachine to catch the ball. As he ran, he was tracked by a videocamera. This had an electronic shutter, set to take images in 2 ms,producing a blur-free image of the fielder. Beyond the fielder wasa wall marked in units of 36 cm. In frame-by-frame replay of thevideo, the position of the back of the fielder's head could beestimated to about 5 cm on the wall. Given the distance betweencamera, fielder, and wall, this uncertainty in measurement corre-sponded to an estimate of the position of the fielder accurate toabout 3 cm. The fielder's position was sampled every 120 ms.The positional estimates from the frame-by-frame analysis weresmoothed with a Hanning window, each position being recalcu-

OnO BowlingV Machine

lated as half of itself plus one quarter of each of its immediateneighbors. The smoothed position estimates were differentiated togive the fielder's velocity.

Position of the ball. The position of the ball in flight was notrecorded on video (except in the final frames when the fielder wasabout to catch it), but it was possible to analytically estimate itsposition throughout the flight. The initial velocity and projectionangle of the ball were known. The distance it traveled was knownbecause its position was recorded on the video as it appearedagainst the structured background just before it was caught. Theflight duration was known (to within 0 ms, the duration of thevideo frame) because a marker appeared on the video at themoment the ball left the bowling machine and the moment whenthe ball was caught was recorded on the video.

These four values were used to compute the trajectory of theball, assuming parabolic flight modified by an aerodynamic dragfactor, proportional to the square of the ball's velocity. The valueof the drag factor was estimated by finding the value that gave thelowest summed mean squared difference between observed andpredicted values of flight distance and flight time. With the best fitvalue for wind resistance, the errors were about 3% in estimatingflight duration and 1% in estimating flight extent. The values weobtained were similar to that given by Daish (1972) for a cricketball. Given drag, initial velocity, and projection angle, it waspossible to calculate the height of the ball and its distance from thebowling machine at any time during the flight. (A detailed accountof the method is given in the article by Dienes and McLeod, 1993.)

Angle of gaze. The initial positions of the fielder and the ballwere known. The position of the fielder after Time t was measuredfrom the video, and the position of the ball after Time t wascalculated as described above. The angle of gaze from the fielderto the ball follows directly.

Ballsflight

Results

TrackingCamera Fielders

start position

Figure 2. The experimental setup as viewed from above. As thefielder ran to catch the ball, he was tracked by the video camera.Given that he was running in a straight line toward or away fromthe bowling machine, his real position could be calculated from theposition that he had reached against the structured background.

Running Speed

The left side of Figure 3 shows six typical examples ofrunning data from 1 fielder. His velocity as he ran to catchthe ball is plotted against time, each curve ending at the timewhen the ball was caught. Each curve is labeled with thedistance he ran, a negative sign indicating that he ranbackward. (All fielders showed qualitatively similar pat-terns. Combining data to show average running patterns, orto compute the variance of the running patterns, is prob-lematic because the fielders ran different distances andpaused for different lengths of time before starting to run.)

Figure 3 demonstrates two effects shown by all fielders.4

First, they were always moving when they caught the ball(except when they had no more than 1-2 m to cover to makethe catch). (Although Michaels and Oudejans, 1992, did notcomment on this effect, it can be seen from their Figure 4that they found the same result. In all of the catches wherethe fielder moved more than about 2 m, she was moving

4 The fielder shown in Figure 3 also showed one effect notshown by all fielders. He always took one or two steps forwardbefore moving backward. This may reflect the fact that in cricket,deep fielders (i.e., ones fielding at some distance from the bat)usually walk toward the batsman as the ball is bowled. However,we do not know why some of our participants did this and othersdid not.


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-0.5

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0.0

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0 1 2 3 0 1 2 3

Time (s)

Figure 3. The left panel shows the fielder's running speed as a function of time as he ran forwardto catch balls 8.8, 5.5, or 2.7 m in front of his starting position or as he ran backward to catch balls2.8, 4.3, or 8.1 m behind his starting position. (He actually started running about 0.5 s after the ballappeared. His velocity is shown as greater than zero slightly earlier as a result of the smoothingalgorithm being applied to the raw position data before the velocity was calculated.) The right panelshows d2(tan a)/dt , where a is the angle of elevation of gaze from the fielder to the ball. The solidline shows the value that we calculated the fielder saw as he ran with the velocity shown on the leftof the figure; the dashed lines show the value that he would have seen if he had run at a constantvelocity, which was either too fast or too slow so that he missed the ball by 2 m.

when she caught the ball.) Thus, the fielder does not run tothe point where the ball will fall and then wait for it butrather runs through the point where the ball will fall at theexact moment that it arrives there. A fielder who knewwhere the ball was going to fall would presumably run tothat point and wait for it to arrive. So, it is possible that thefielder does not know where the ball wil l fall when running.

This possibility suggests a solution to the problem, illus-trated by Figure 1, that when the fielder starts to run, thereappears to be insufficient information to work out where theball will land. This lack of information is paradoxical if thefielder is assumed to know where the ball will land whenstarting to run. However, if it turns out that the fielder doesnot know where the ball will land, the problem disappears.There may be sufficient information in the first few hundred

milliseconds of the ball's flight to tell the fielder in whichdirection to start running, even though there is not enough totell where or when the ball will land.

The second point made by Figure 3 is that the runningpatterns for different distances had nothing in common.Long runs involved continuous acceleration, medium dis-tances involved acceleration and then constant velocity, andshort distances involved acceleration and then deceleration.The requirement of Chapman's (1968) analysis, that fieldersshould run at constant velocity, did not hold. A variant ofChapman's proposal that might allow for this has been putforward by Babler and Dannemiller (1993). They suggestedthat the fielder starts running at the constant velocity thatzeros optic acceleration, as Chapman suggested. When,owing to the nonparabolic flight, optic acceleration starts to


increase or decrease, the fielder finds a new velocity thatzeros it, and so on, repeatedly throughout the flight. Theidea, in effect, is that a nonparabolic flight could be approx-imated by a series of roughly parabolic sections. Thus, evena fielder following Chapman's method might not run at thesame speed throughout the catch. Although Babler andDannemiller's proposal seems intuitively reasonable, theygave no proof of the effectiveness of this algorithm. Nordoes it resolve the problem, inherent in Chapman's strategy,of how the fielder finds the constant running speed thatresults in zero optic acceleration. Whether the strategy couldwork in principle, it is difficult to see either of the catchesat the longest distance, with continuous accelerationthroughout the flight, offering support even for this modi-fied version of Chapman's proposal.

Angle of Gaze

The right side of Figure 3 shows the value of d2(tan a)/dt2

(where a is the angle of elevation of gaze from the fielderto the ball) that the fielder would have seen as he watchedthe ball during each run up to 240 ms before the catch wasmade.5 It appears that he waited for about 0.5 s and thenstarted to run, accelerating until he reached a speed whered2(tan a)/dt2 = 0. Then, he modulated his speed up to thepoint of catching the ball. If he was running forward, he ranfaster if d2(tan a)/dt2 became negative and more slowly if itbecame positive, and if he was running backward, viceversa.

Clearly, d2(tan a)/dt2 is maintained close to zero, but is itclose enough to ensure that the ball is caught? This can beassessed by considering what the value would be if thefielder just failed to reach the ball. Given that arm reach isabout 1 m and allowing for a jump or lunge as the ball iscaught, the fielder would just be unable to catch the ball ifhe was about 2 m away from the ball when it reachedcatching height. The dashed lines show the value of d2(tana)/dt2 that the fielder would have observed had he run at aconstant speed that would have taken him to a point either2 m short of or 2 m beyond the place where the ball wouldfall. It can be seen that in every flight the value of d2(tana)/dt2 actually experienced by the fielder would take him toless than 2 m from the place where the ball would fall. Thedeviations of d2(tan a)/dt2 from zero were insufficient toprevent the fielder from intercepting the ball.

To test the claim that d2(tan a)/dt2 = 0, we plottedregression lines of d2(tan a)/dt2 against time, from the timewhen the fielder started running until 240 ms before hemade the catch. For a random sample of 15 successfulcatches by the fielder whose data are shown in Figure 3, themedians of the absolute values of the intercepts and theslopes of these regression lines (i.e., taking the median ofthe absolute value and ignoring the sign) were 0.02 s~2

(signed range from -0.07 to 0.07) and 0.02 s~3 (signedrange from -0.04 to 0.10), respectively. In only one flightwas the value of either slope or intercept reliably different(at the 5% level) from zero. A random sample of 27 suc-cessful catches from the other fielders showed a similar

result: The median values of the absolute intercepts andslopes were 0.04 s~2 (signed range from -0.08 to 0.09) and0.04 s~3 (signed range from —0.04 to 0.09), respectively.Only 2 of the catches gave either a slope or an interceptreliably different from zero.

Experiment 2

Experiment 1 showed that when the fielder had less far torun he ran more slowly, rather than running to the pointwhere the ball would fall and waiting for it (see the running-speed data in Figure 3). If he knew where to go, runningmore slowly and arriving just in time to catch the ball wouldseem a pointlessly risky strategy. Why not go to the rightplace and wait? However, if the fielder does not knowwhere the ball will land but is following a strategy that willget him to the right place at the right time, it is inevitablethat he will run more slowly if he has more time. Experi-ment 2 was a direct test of this possibility.

Method

The projection angle of the ball was increased to 64°. With aninitial projection velocity of 24 m/s, it now fell about 36 m fromthe bowling machine, a distance similar to the balls projected at45° and 20 m/s in Experiment 1. However, because the trajectorywas higher, the ball took longer to get there. If the fielder knewwhere he was going, he could go there and wait for the ball on thehigher, longer trajectory. But if he was following a strategy thatwould lead him to arrive at the same time as the ball, he would runmore slowly.

Results

Running Speed

Figure 4 (upper panel) shows the fielder's speed for sixdifferent catches as he ran 8-10 m to catch the balls on thetwo different trajectories. The upper curve shows his meanspeed for three catches when the ball was projected at 45°;the lower curve shows his mean speed for three catcheswhen the ball was projected at 64°. The bars show the rangeof speed over the three runs. It can be seen that the sepa-ration of the curves representing the means is an accuratereflection of the individual catches because there is nooverlap between me individual curves from the two groups.The curves end at the point where the fielder made thecatch. In both cases, he arrived at the point where the ballfell at the same time as the ball (i.e., he had a positivevelocity at the moment that he made the catch). With the

5 Like Michaels and Oudejans (1992), we usually found a cat-astrophic jump in d2(tan a)/dt2 over the last two data points. Thesudden change in a that gave rise to this effect was caused, at leastin part, by the fact that the ball was not caught at the point that wastaken to represent the origin of the fielder's angle of gaze. Thesudden change in a may well have had some role in the terminalreach adjustment immediately before the catch, but it occurred toolate to be relevant to the question of how the fielder got closeenough to the ball to make the catch, so we ignored it.


2

Timefc)

4-2

are

-2

Figure 4. The fielder ran to catch balls landing 8-10 m in frontof him. The ball had an initial projection angle of 45° (flight timeabout 3 s) or 64° (flight time about 4 s). The top panel shows thefielder's running speed as a function of time as he ran to catch theball. Each curve is the mean of three runs. The bars show the rangeof velocities across the three runs. In the bottom panel, the solidlines show d2(tan a)/dt2 for each run, and the dashed lines showwhat the value would have been if he had run at constant velocityto a point 2 m short of or 2 m beyond the point where the ball fell.

longer trajectory, he ran more slowly. This finding supportsthe conclusion of Experiment 1: The strategy he used gothim to the right place at the right time. However, it does notappear to tell him where that place is in advance.

Angle of Gaze

Figure 4 (lower panel) plots the value of d2(tan a)/dt2 foreach run. The solid lines are the values of d2(tan a)/dt2 thatthe fielder would have seen as he ran (averaged over thethree flights). The dashed lines show what he would haveseen if he had run at constant velocity to a point 2 m shortof or 2 m beyond the place where the ball fell.

The upper panel of Figure 4 shows that two quite differentrunning patterns were produced to get the fielder to thesame place when the ball's trajectory was changed. Whatthey have in common is that the fielder ran at a speed thatkept d2(tan a)/dt2 close to zero. Regression lines of d2(tana)/dt2 against time, plotted for the individual catches, showonly one catch for which either intercept or slope wasreliably different from zero. For the 45° projection angle,the medians of the absolute values of the intercepts and theslopes were 0.04 s"2 (range from -0.07 to 0.04) and 0.02s~J (range from 0.02 to 0.04), respectively. For the 64°projection angle, the medians of the absolute values of theintercepts and the slopes were 0.07 s~2 (range from —0.07to 0.09) and 0.04 s~* (range from -0.04 to 0.05), respec-tively.

Conclusion

The fact that the fielders did not use spare time to run tothe place where the ball would fall and wait suggests thatthey did not know where it would fall. However, the exper-iments did not directly test the fielders' knowledge of wherethe ball would fall. It can be concluded that the algorithmfielders use to intercept the ball is one that ensures theyarrive at the right place at the right time but does not tellthem where or when that is. Whether fielders know wherethe ball will land but choose not to use this information asthey run to catch it is a possibility that awaits furtherexperimentation.

Missing the Ball

We claim that fielders use the sign of d2(tan a)/dt2 as theinput to a servo that controls running speed. When runningbackward, they speed up when it is positive and slow downwhen it is negative, and when running forward, vice versa.This strategy is illustrated by seeing what happens when afielder fails to run fast enough to catch the ball. Figure 5compares two catches where the ball landed in roughly thesame place, about 6 m behind the starting position of thefielder. In one case (open circles), he successfully caughtthe ball; in the other (filled circles), he started runningbackward too slowly, and the ball went over his head, justout of reach of his outstretched hand.

The upper panel of Figure 5 shows his velocity. For thesuccessful catch, he initially accelerated backward, eventu-ally decelerated, and maintained an approximately constantvelocity until he made the catch. In the unsuccessful case,he was slower to start and accelerated more slowly but


-0

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a.-4

0.2

"3-§0.1

0-o

-0.12

Time (s)

Figure 5. Velocity and d2(tan a)/dt2, as in Figure 3, for twocatches where the fielder ran backward. Open circles represent thathe caught the ball; filled circles represent that the ball went overhis head.

continued to accelerate throughout the flight of the ball.Why did he produce these different running patterns?

The lower panel of Figure 5 shows d2(tan a)/dt2 for eachflight. In both cases, the value started positive (because theball was going over the fielder's head) and increased (be-cause the fielder was initially stationary). Once the fielderstarted to run in the appropriate direction, the value camedown. For the successful catch (open circles), it reachedzero, so he stopped accelerating. For the unsuccessful catch(filled circles), d2(tan a)/dt2 remained positive throughoutthe flight, so the fielder continued to accelerate but to noavail. He started too late and could not run fast enough tointercept the ball.

General Discussion

We manipulated the time that a fielder had to run for acatch in two ways. We made the ball fall nearer to or fartherfrom him (Experiment 1); we made it land at the same place,but it took longer to get there (Experiment 2). In bothexperiments there were apparently complex changes in thefielder's running speed (see Figures 3 and 4). But in allcases one thing remained constant: He ran at a speed thatkept the acceleration of the tangent of the angle of elevationof gaze close to zero. This result was predicted by Chapman(1968). But we know that Chapman's analysis cannot becorrect. First, it is based on the information available fromwatching a parabolic flight. The balls in this experimentwere thrown sufficiently fast to have departed considerably

from parabolic flight.6 Second, Chapman's algorithm as-sumes that fielders run at constant velocity. Figures 3 and 4show that typically they do not do so.

A possible resolution is Babler and Dannemiller's (1993)suggestion that these two problems are linked. They pro-posed that as the flight departs from parabolic, the fieldersadjust their running velocity, finding a series of new valuesof constant velocity throughout the flight successively ze-roing out optic acceleration. But a fundamental problem stillremains. To discover that Chapman's (1968) strategy worksrequires people to view parabolic flights (which they never,in fact, experience) while running at one particular constantvelocity (or set of constant velocities, if we adopt Babler &Dannemiller's, 1993, argument). The value of the constantvelocity (or set of velocities) would be different for everyflight they experienced. The implausibility of anyone everdiscovering that Chapman's algorithm led to interception,coupled with the fact that most children learn to catch justby watching balls in flight, suggests the need for a differentapproach.

How to Intercept a Ball

Consider Figure 6a. A ball is falling, watched by a fielder.The angle of elevation of gaze from the fielder to the ball isa. The height of the ball above the ground is y, and thehorizontal distance from the fielder to the ball is x. Therequirement for intercepting the ball before it hits theground is simple: As y —> 0, x —> 0. That is, as the ball dropsto the ground, the fielder reaches the place where it drops.This is illustrated in Figures 6b and 6c, where the fieldercloses in on the ball (x —» 0) as it falls (y —> 0). If both x andy —» 0 together, a will always be positive but less than 90°,that is, 0° < a < 90°. If the fielder fails to intercept the ball,one of two things must happen: Either it falls in front ofhim, in which case a < 0° (Figure 6d) or it goes over hishead, in which case a > 90° (Figure 6e).7

The conditions for intercepting or, alternatively, failing toreach the ball are surprisingly simple: If the fielder runs ata speed that ensures that the angle of gaze is greater than 0°but less than 90° throughout the flight, the ball will beintercepted. If the angle of gaze reaches either 0° or 90°, the

6 For example, the ball projected at 64° and 24 m/s in Experi-ment 2 traveled 36 m. In parabolic flight, it would have traveledabout 46 m.

7 Strictly, this is true only if balls can only be caught just in frontof the eyes—the point from which the angle of elevation of gazeis measured. Of course, fielders can stretch their arms forward andcatch a ball a few feet in front of them, despite the fact that a hasgone to 0°, or catch one going just over their head, when a willhave reached 90°. However, in cricket at least, fielders prefer tocatch a ball that has been hit high in the air just in front of theireyes (see, e.g., Richards & Murphy, 1988, p. 127). They stretchtheir arms out to catch the ball as a last moment adjustment onlyif they have failed to get to the right place. We based our analysison the assumption that fielders are endeavoring to run to theoptimum place for catching, realizing that some other algorithmalso exists to control the arm movements that allow for a correc-tion just as the ball arrives if they fail to reach the ideal point.


0<a<90°

0<a<90

Q!=0

a=90

Figure 6. The conditions for intercepting or missing the ball, (a) a is the angle of elevation of gazeas the fielder watched the ball. If the fielder ran at a speed that kept a positive but less than 90° asthe ball fell (b and c), it would be intercepted. If a fell to 0° or reached 90°, the ball would not beintercepted because it would have fallen in front (d) or gone over the fielder's head (e).


ball will be missed (with the exception mentioned above ofa ball within reach of an outstretched hand). This is illus-trated in Figure 7, which shows the angle of gaze for thefielder in Figure 1 watching the flight of the balls on thethree trajectories shown there. (Figure 7 shows the angle ofgaze for a stationary fielder. If the fielder ran to catch theball, unsuccessfully in the case of the ball landing in front ofor behind him or successfully in the case of the ball landingin his hands, the picture would be qualitatively similar.) Theangle of elevation of gaze, a, starts at zero in each case andincreases as the fielder watches the ball rising in the air. Forthe ball that will land in his hands, a continues to increasethroughout the flight. But the rate of increase slows down asthe flight progresses, and a never reaches 90°. For the ballthat will fall in front of him, a reaches a maximum, starts todecrease, and accelerates back toward zero. For the ball thatwil l go over his head, a accelerates throughout the flight,reaching 90° as the ball passes over his head. Although theexact way that a changes throughout the flights will , ofcourse, vary with angle and velocity of projection and thewind resistance of the object, this general pattern is shownfor all flights. For any flight that the fielder misses, areaches either 0° or 90°; for any flight that is intercepted,0° < a < 90° throughout the flight.

Why Running so That (f(tan a)/df = 0 Results inthe Ball Being Intercepted

The reason why running at a speed that keeps d2(tana)/dt2 = 0 leads to interception can be understood byconsidering the usual case where the fielder starts to runwhile the angle of elevation of gaze is increasing; a, andthus tan a, will be positive and increasing when the fielderstarts. If the fielder runs at a speed that keeps d2(tan a)/dt2

= 0, tan a must be positive and finite at the end of the flight(i.e., 0 < tan a < oo). Because tan 0° is 0 and tan 90° is oo,it follows that a will lie between 0° and 90° at the end of theflight. But this is the condition for intercepting the ball.Therefore, if the fielder can run fast enough to keep d2(tana)/dt2 = 0, the ball will be intercepted.8

Chapman and d2(tan a)/df

Chapman (1968) demonstrated a curious geometrical re-lationship. If a fielder runs toward the place where a ball inparabolic flight will fall at the constant velocity that willcause him or her to arrive at the same time as the ball, d(tana)/dt will be constant throughout the run. Because, bydefinition, Chapman's fielder successfully intercepts theball, this observation has sometimes been seen as offering asolution to the problem of how to catch the ball. The factthat Michaels and Oudejans (1992; and now we) haveshown that d(tan a)/dt is constant, that is, d2(tan a)/dt2 = 0,as fielders run is seen as support for this belief.

Chapman's (1968) observation is geometrically correct,but it does not offer a solution to the problem facing thefielder. To run at constant velocity to the interception pointrequires the fielder to know, when he or she starts, what

distance must be run in what length of time. Running atconstant velocity thus requires the fielder to know whereand when the ball will land. So Chapman's observation doesnot lead to a solution to the problem of intercepting the ballbecause it requires the fielder to know the answer (i.e.,where and when the ball will land) before it can be imple-mented.

An alternative way of construing Chapman's (1968) ob-servation into a solution of the problem facing the fielder isto turn it round. If the fielder runs so as to keep d(tan a)/dtconstant at one particular value, velocity will turn out to bethe one constant velocity that produces interception. Theproblem with this is that the crucial value of d(tan a)/dtwould be different for every ball trajectory and every start-ing position. There is no reason to believe that the fieldercould know all these values. (There is also the empiricalproblem for this approach that people do not run at constantvelocity!)

Although we agree with Chapman (1968; and withMichaels & Oudejans, 1992) that a crucial part of thefielder's strategy involves keeping d2(tan a)/dt2 at zero, ourinterpretation of what the fielder is doing is entirely differ-ent from theirs. We believe that keeping d(tan a)/dt constantleads to interception because it keeps a between 0° and 90°,not because it produces the one constant running velocitythat will produce interception. Chapman showed that inter-ception would occur if the fielder kept the one value of d(tana)/dt constant that produces constant running speed. In fact,a fielder who keeps any value of d(tan a)/dt constant willintercept the ball (see Dienes & McLeod, 1993), but onlythe "Chapman" value for any given flight will result in aconstant running velocity.

Once one realizes that any constant value of d(tan a)/dtwil l produce interception, the problem for the fielder, in-herent in Chapman's (1968) strategy, of knowing the crucialvalue of d(tan a)/dt disappears. The fielder, having sightedthe ball for long enough to have generated a value of d(tana)/dt (which will be different for different trajectories),simply has to run at a speed that keeps it constant. Thisspeed will usually vary during the run. If the fielder waits alittl e longer before starting to run, the initial value of d(tana)/dt will change, and a different running pattern will result.But the ball will still be intercepted if the fielder can run fastenough to keep d(tan a)/dt constant.

Alternative Strategies

Interception will occur provided the fielder ensures that0° < a < 90° throughout the ball's flight. The strategy thatskilled fielders appear to use, keeping d2(tan a)/dt2 at zero,has been shown to be a very effective strategy for doing this.It works for any trajectory and whatever the time the fielderstarts running, provided the ball has a horizontal velocity

8 Surprisingly, this strategy usually leads to interception even ifd(tan a)/dt < 0 (i.e., the ball is already falling) when the fielderstarts to run. Proof of this and an analysis of the conditions whenthe strategy wil l not work can be found in Dienes and McLeod(1993).

540 McLEOD AND DffiNES

90

80

70

60

50

! 40

30

20

10

.24 .48 .72 .96 1.20 1.44 1.68 1.92 2.16 2.40 2.64 2.88 3.12Tlme(s)

Figure 7. How the angle of elevation of gaze, a, varied with time as the stationary fielder inFigure 1 watched the flights of the ball falling in front (1), at (2), or going overhead (3).

component toward the fielder (Dienes & McLeod, 1993).However, it might seem unnecessarily complex. We exam-'ined two strategies that might seem simpler and thus moreplausible but showed that they were not as effective.

Constant a

The simplest strategy for ensuring that a lies between 0°and 90° is to maintain a constant a. Maintaining a constanta is equivalent to keeping d(tan or)/dt rather than d2(tana)/dt2 at zero.

In games like cricket and baseball, the ball is seldom inthe air for more than a few seconds. So one limit on theability of fielders to catch the ball is set by how far they canrun while the ball is in the air. The sooner fielders startrunning in the right direction, the farther they will be able torun in the time available. Thus, the most effective strategyis the one that gets the fielder running in the right direction,at the highest speed, soonest.

Figure 8 shows the flight of a ball that will fall in front ofthe fielder. To intercept it, the fielder must run forward. Afielder who tries to ensure interception by maintaining aconstant angle of gaze (see Figure 8a) will start by runningin the wrong direction. As the ball descends, the fielder willthen have to change direction, recovering lost ground beforestarting to make any progress from the original startingposition to the place where the ball will land. Obviously amore effective strategy is one that sends the fielder in thecorrect direction immediately. To do this, the angle of gazemust be allowed to increase (see Figure 8b).

The same argument applies if the ball is going to land

behind the fielder. A fielder who maintains a constant angleof gaze will pass the place where the ball will land and haveto run back to it (see Figure 9a). In contrast, a fielder whoallows the angle of gaze to increase throughout the flightcan arrive at the place to make the catch without overshoot-ing it (see Figure 9b).

Constant d(a)/dt

The analysis described above shows the importance ofallowing a to increase. One way to accomplish this is to runso that d(a)/dt remains constant, that is, keeping d2(a)/dt2

rather than d2(tan a)/dt2 at zero.The problem with letting a increase is the danger that it

wil l exceed 90°. Dienes and McLeod (1993) showed, forexample, that allowing a to increase at constant velocity ledto fielders running back too slowly to catch balls that weregoing over their heads on trajectories that we know, exper-imentally, skilled fielders would catch. Keeping d2(a)/dt2 atzero allows a to exceed 90° before the fielder reaches theplace where the ball can be caught. Keeping d2(tan a)/dt2 atzero also allows a to increase. However, because tan agrows more quickly than a, the rate of increase slows downas a increases. Because tan a grows more and more quicklyas a approaches 90°, keeping d2(tan a)/dt2 at zero ensuresthat the rate of increase of a slows more and more as aincreases. Thus, provided the fielder can run fast enough tokeep d2(tan a)/dt2 at zero, this strategy ensures that a willnever exceed 90°.

The way that the strategy followed by expert fieldersallows a to increase but slows down the rate of increase as


Figure 8. Two strategies for interception when the fielder should run forward, (a) Keeping theangle of elevation of gaze constant is a poor strategy because it can send the fielder in the wrongdirection, (b) Allowing the angle of gaze to increase ensures that the fielder moves in the rightdirection.

a increases can be seen by plotting the value of a againsttime for the six catches shown in Figure 3. This is shown inFigure 10. The reduction in the rate of increase of a as aincreases is demonstrated by the highly significant quadraticcomponent to multiple regressions of a on t (p < .001 foreach catch). This shows that fielders do not keep d(a)/dtconstant as they run. Thus, as complex as the strategy ofkeeping d2(tan a)/dt2 at zero might seem, there are goodreasons for both elements of it to have evolved.

What Do Children Learn?

Children who are learning to catch start by watching ballsthrown toward them while standing still. Since 0° < a <

90° will be true of all flights that land in their arms, and onlythose flights, it would not be surprising if they discoveredthe importance of this relationship. It seems reasonable toassume that they might try to use this fact when they haveto start running for the ball. That is, they will try to keep0° < a < 90° as they run.

But why should they discover that maintaining d2(tana)/dt2 at zero is an efficient way of keeping 0° < a < 90°?Extending the original observations of Chapman (1968)about watching parabolic flights, Dienes and McLeod(1993) showed that for a stationary fielder, at any point inthe ball's flight and whatever its trajectory (i.e., independentof the effects of wind resistance), the sign of d2(tan a)/dt2 isalmost always negative if the ball will land in front of the


Figure 9. Two strategies for interception when the fielder is running backward, (a) Keeping theangle of elevation of gaze constant is an inefficient strategy because it will send the fielder past thepoint where the catch can be made, (b) Allowing the angle of gaze to increase avoids this problem.

observer and almost always positive if it will land behindthe observer.9 So the fact that d2(tan a)/dt2 is correlatedwith the catchability of a ball flight is available to thestationary child. Whether mere exposure is sufficient toallow acquisition of this fact is a question that remains to beanswered.

Are People Really Computing d2(tan a

Our data support those of Michaels and Oudejans (1992)in suggesting that competent catchers have discovered theeffectiveness of keeping d2(tan a)/dt2 at zero as the basis foran interception strategy. Of course, our data do not indicatehow the computational problem of keeping d2(tan a)/dt2 atzero is solved. Skepticism about the conclusion might stemfrom the feeling that d2(tan a)/dt2 does not seem a partic-ularly likely quantity for the nervous system to represent.

Todd (1981) showed that if participants knew the actualsize of the ball and the acceleration due to gravity, zeroingan expression involving the second power of the optic sizeof the ball was equivalent to maintaining d2(tan a)/dt2 atzero in vacuo. Direct extension of Todd's analysis shows

that the wind resistance problem can be overcome by usingan expression involving the third power of optic size and thesquare of its first derivative. However, given that fieldersstart running almost immediately to catch balls thrown fromdistances of more than 50 m, an approach requiring acutesensitivity to optic size seems implausible. It is possible thatfielders do not compute tan a at all. If they were to reducea maintained value of d(a)/dt in a systematic way as aincreased (as described in Dienes & McLeod, 1993), thiswould keep d2(tan a)/dt2 at zero.

Another problem with the strategy of maintaining d2(tana)/dt2 at zero is that the visual system is generally notparticularly sensitive to acceleration. However, both Bablerand Dannemiller (1993) and Tresilian (1995) have shownthat the performance reported by us and by Michaels andOudejans (1992) can be achieved by a system with thelimitations of the human visual system.

9 The limitations that lead to the qualifications almost and usu-ally are described in detail in Dienes and McLeod (1993).


80

60

40

20

t (s)

Figure 10. Angle of gaze as a function of time for the sixcatches shown in Figure 3. The numerical label shows how far (inmeters) the catcher ran to make the catch. A negative sign indicatesthat he ran backward.

References

Babler, T. G., & Dannemiller, J. L. (1993). Role of image accel-eration in judging landing location of free-falling projectiles.

Journal of Experimental Psychology: Human Perception andPerformance, 19, 15-31.

Brancazio, P. (1985). Looking into Chapman's homer: The physicsof judging a flyball. American Journal of Physics, 53, 849-855.

Chapman, S. (1968). Catching a baseball. American Journal ofPhysics, 36, 868-870.

Daish, C. (1972). The physics of ball games. London: The EnglishUniversities Press.

Dienes, Z., & McLeod, P. (1993). How to catch a cricket ball.Perception, 22, 1427-1439.

Michaels, C., & Oudejans, R. (1992). The optics and actions ofcatching fly balls: Zeroing out optic acceleration. EcologicalPsychology, 4, 199-222.

Regan, D. (1993). Binocular correlates of the direction of motionin depth. Vision Research, 33, 2359-2360.

Regan, D., Beverley, K., & Cynader, M. (1979, July). The visualperception of motion in depth. Scientific American, 241, 122-133.

Regan, D., & Kanshal, S. (1994). Monocular discrimination of thedirection of motion in depth. Vision Research, 34, 163-177.

Richards, V., & Murphy, P. (1988). Viv Richards' cricket master-class. London: Queen Anne Press.

Todd, J. (1981). Visual information about moving objects. Journalof Experimental Psychology: Human Perception and Perfor-mance, 7, 795-810.

Tresilian, J. (1995). Study of a servo-control strategy for projectileinterception. Quarterly Journal of Experimental Psychology, 48,688-715.

Received November 3, 1993Revision received February 13, 1995

Accepted March 21, 1995

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